@SillyGoose I know you are somewhat identifying with the OP question. The answer is that the direction you are going for is actually not the way to go. What instead is happening is that you can prove that if you get all the irreps of SU(2), then you will have already solved the problem in general. Then you study the properties of those that can, or cannot, be represented by x and p, and you see that only odd dimensional ones work
So, the answers as written actually answered the question, by sending the OP into the correct direction to solve it.
(And the OP also states that if you want to see the proof, go look up the book)
Well I mention my problem with the book mentioned in the post: Hall starts at the outset looking for genuine (not projective) representations of SO(3) for orbital angular momentum @naturallyInconsistent
It does matter because by not considering projective representations you make the entire thing trivial in that only odd dimension irreps are allowed for genuine reps of SO(3)
you are ignoring the possibility of the even dimension irreps when there's no justification (that i can find in the book) for doing so
when it is precisely ignoring projective reps that would lead one to an erroneous theory of spin
So you are making an assumption that we actually know to not hold for angular momentum as a whole without justifying said assumption (i.e. ruling out projective reps for orbital angular momentum)
@naturallyInconsistent Now this explanation I can buy and I have bought it. But I am looking for a way to see this without deferring to teasing out the lie algebras; if that is genuinely not possible then I can accept that. but i find it hard to believe that even a qualitative story describing the distinction between the projective (or not projective as it seems for orbital angular momentum) representations of $SO(3)$ for spin versus orbital angular momentum using representation theory and not referring to
@SillyGoose The OP was saying that they found a proof that the even dimensional part cannot be represented by position and momentum. That's what I was bringing to your attention
@Mr.Feynman oh? not a full proof? well, better than nothing
Well I am okay with Hall's proposition there. My problem is that he is considering only genuine representations. projective representations must allow even dimension irreps (as in the spin irreps of SO(3) )
I just do not think that it is an important problem enough to be spending your obscene amounts of effort into. It is a nice problem to solve, but not going to be show-stopping either direction
it is a really irksome problem D: and idk it would be nice to fill in what seems like a genuine gap in the pedagogical presentation of textbook non rel quantum
@SillyGoose I fail to see how this is a pedagogical problem. I understand that the textbook quantum theory is not presenting enough to quell questions, but no textbook should be solving them all. It would be just too big a book.
I think having done Feynman lectures is really good, because he really milks the Stern-Gerlach experiment to tease out how spin-half can be totally extracted out of it.
@naturallyInconsistent I will try to distill my thoughts more to answer this question and give a meaningful response
@Mr.Feynman Hm perhaps; but if one is able to deduce even if heuristically what i would like to deduce without deferring to the algebras, it would make for a much prettier story :P
I'm not trying to write for mathematicians, so I will not be going straight into irreps and shit, but as a theorist myself, I would not be shying away from the maths
@naturallyInconsistent I think I personally hold the impression that the theory of angular momentum occupies a substantial part (a chapter in Sakurai at least) of a first UG course in Quantum Mechanics. This is probably a student's first exposure to Lie and Representation theoretic concepts as they relate to physics even if it is just all going on under the hood. But in any case, the theory of angular momentum qua the theory of angular momentum seems has some importance.
Now, for a first UG course in QM perhaps it is sufficient to do use commutation relations derived from Stern-Gerlach and then essentially build up the whole general theory using these commutation relations.
However, this is a supremely cumbersome picture to have of angular momentum. Should one do this process every time they would like to “add angular momenta”? No, it is more natural to understand the representation theoretic description of angular momentum for these purposes.
Moreover, I personally think it is more natural to understand angular momentum in general via the representation theoretic description. From this representation theoretic point of view it is obvious how one should “add angular momenta”.
Now to understand the representation theoretic picture of angular momentum, all one has to do is understand just how exactly is angular momentum related to representations of SO(3). This in turn leads to the question of how spin and orbital angular momentum separately are related to representations of SO(3) as they are the two manifestations of angular momentum.
And it would have very little to do with how it would actually transform in space, unless the author and the students actually put in much more effort to derive those.
Oh, no, I am already assuming that they understand linear algebra enough. I am not talking about USA students
whereas students can follow through Feynman's tedious calculation and see how spin half does, and then follow through how to add two spin halfs and see that they break into spin zero and spin 1
basically, the problems in QM for experimenters will not need to go beyond 5 such combinations. They will not be needing the full apparatus of rep theory
I will still cover some bits of that, but I will not be covering everything
It is the whole, students want to quickly see the rewards from putting in maths effort
Which is kinda sad on its own, but still, understandable
@naturallyInconsistent Hm I agree with this statement most. It is sad. I cannot comprehend how anyone can understand Quantum (other than in an operational manner) without investing some time and interest into the mathematical machinery running under the hood. But yeah not everyone is interested in Quantum qua quantum who takes quantum to begin with I suppose.
@SillyGoose R u asking y we dont go out looking for projective representations of $SO3$ on $L^2(R^3)$? I mean, y we r satisfied with the $L_x$, $L_y$, $L_z$ operators on $L^2(R^3)$ and y we dont look for other operators on $L^2(R^3)$ which satisfy the $su2$ algebra and possibly have 1/2 integer eigenvalues
I want 2 say that the $L_i$ are Hermitian, so their eigenspace spans the full $L^2(R^3)$ space becuz of spectral theorem. So there r no more eigenvectors and eigenvalues that r needed in spanning the Hilbert space
But more importantly, I think that any othr representation of $su2$ on $L^2(R^3)$ shud b unitarily equivalent to $L_x , L_y, L_z$ becuz the commutation relations r the same. But idk how to prove this
So becuz of this unitary equivalence, any othr rep of $su2$ on $L^2(R^3)$ shud hav the same eigenspectrum as $L_x, L_y, L_z$
But this unitary equivalence shud b proved. It's not to b taken 4 granted
@RyderRude this is essentially one of the questions. the more important question is in an arbitrary hilbert space how does one partition it into a tensor product of $L^2$ and a finite dimensional hilbert space such that the finite dimensional space coincides with containing the spin degrees of freedom withoutt specifying this fact at the outset
Oh ur main question seems very complicated. How would we know how to partition it
The way it is approached in courses is using quantisation of a classical theory. Replacing the phase space functions with operators yields us the usual $L_i$ operators on $L^2(R^3)$. And then we latter add spin on top of it to model electrons
But u r looking for a very bottom-up approach where we r just given a Hilbert space and we have to work out the representations of $su2$
Tbf the partitioning isnt even necessary. The reprentations of $J_i=L_i +S_i$ and $J^2$ intermingle the finite and infinite degrees of freedom.
Ah, yeh.. once I was lying down reading a book... and I laughed 'cause there was something funny.. suddenly the laughter appeared to be shaking the bed more than I expected it to... turned out it was an earthquake!
Can someone help me understand why do we need $dg$ at all in the way he arrives to it? It maps a vector to itself, so why would I compose $(L_{g^{-1}})_*$ with it instead of just having $\Omega=(L_{g^{-1}})_*$?
@Slereah for Cech you need to say of which sheaf, but the constant sheaf for the reals has as Cech the de Rham cohomology, cf math.stackexchange.com/q/673518/143136
Lol, after searching MSE and MO, the origin of my question was here on PSE. The motivation isn't compelling yet but at least my question has been asked somewhere
I mean, the differential of the left translation is a $\mathfrak{g}$-valued differential form itself
I can't seem to find any info about this, but does anyone have any insights about taking up a theory position in an experimental research group (condensed matter)? I am slightly worried about not being able to pursue research as freely as I would like.
